Pythagorean triangles of equal areas

Malvina Baica

International Journal of Mathematics and Mathematical Sciences, 1988, vol. 11, 1-12

Abstract:

The main intent in this paper is to find triples of Rational Pythagorean Triangles (abbr. RPT) having equal areas. A new method of solving a 2 + a b + b 2 = c 2 is to set a = y − 1 , b = y + 1 , y ∈ N − { 0 , 1 } and get Pell's equation c 2 − 3 y 2 = 1 . To solve a 2 − a b − b 2 = c 2 , we set a = 1 2 ( y + 1 ) , b = y − 1 , y ≥ 2 , y ∈ N and get a corresponding Pell's equation. The infinite number of solutions in Pell's equation gives rise to an infinity of solutions to a 2 ± a b + b 2 = c 2 . From this fact the following theorems are proved.

Theorem 1 Let c 2 = a 2 + a b + b 2 , a + b > c > b > a > 0 , then the three RPT-s formed by ( c , a ) , ( c , b ) , ( a + b , c ) have the same area S = a b c ( b + a ) and there are infinitely many such triples of RPT.

Theorem 2 Let c 2 = a 2 − a b + b 2 , b > c > a > 0 , then the three RPT-s formed by ( b , c ) , ( c , a ) , ( c , b − a ) have the same area S = a b c ( b − a ) and there are infinitely many such triples of RPT.

Date: 1988
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:251943

DOI: 10.1155/S0161171288000948

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